464 Mr. Sylvester's Analytical Development 



. . . + b* - a 2 = 

 And in like manner interchanging b t y, m with r, *, n 



+ c 2 — tf 2 = 0. 

 Hence if ( — — ) ( ^ """"^ ) De tne two systems of values 



of i, Athene .JiZi.jfY ij-i) 

 ^ a? \X ar, X V\X #„ X a?,,/ 



are the two lines of vibration required. 



Proposition 2. 

 By last proposition it appears that 



Z£* = c *~ a * ...... ^ y 



m § m m ft* — c 2 • v •/ 



and ^ = j^L£ w 



% . y,y„ + z , z „ - g9 -^ 3 _ _ j 



#, #„ W- — c* 



And .*. the two lines of vibration are perpendicular to each 

 other. 



N.B. = ns (c) and (d) must not be overlooked. 



Proposition 3. 



A line of vibration is given (i. e. ^1 ~L are given ) and 



the position of the front is to be determined. 



Let Ix + my +'nz = Obe the front required, 

 then Ixt H- m y t + n n\ = 0, 



and (6 2 - c 2 ) — + (c 2 -*j - + ^"=T 2 . -,= 0. 

 Eliminating (w) we get 



V 5 »i / 



